Y is a subspace of a X which is a topology

Let Y be a subspace of X. If A is closed in Y and Y is closed in X, then A is closed in X.

First are there 2 cases to consider? (1) A is a proper subset of Y (2) $\displaystyle A=Y\cap U$ or does this not matter?

Anyways, all I have is this:

Since A is closed in Y, $\displaystyle Y-A$ is open in Y. Likewise, $\displaystyle X-Y$ is open in X.

I am not sure what to do next.

Re: Y is a subspace of a X which is a topology

Quote:

Originally Posted by

**dwsmith** Let Y be a subspace of X. If A is closed in Y and Y is closed in X, then A is closed in X.

First are there 2 cases to consider? (1) A is a proper subset of Y (2) $\displaystyle A=Y\cap U$ or does this not matter?

Anyways, all I have is this:

Since A is closed in Y, $\displaystyle Y-A$ is open in Y. Likewise, $\displaystyle X-Y$ is open in X.

I am not sure what to do next.

By definition that $\displaystyle A$ is closed in $\displaystyle Y$ you have that $\displaystyle A=Y\cap C$ for some closed subset $\displaystyle C$ of $\displaystyle X$....so.

Re: Y is a subspace of a X which is a topology

Quote:

Originally Posted by

**Drexel28** By definition that $\displaystyle A$ is closed in $\displaystyle Y$ you have that $\displaystyle A=Y\cap C$ for some closed subset $\displaystyle C$ of $\displaystyle X$....so.

I don't know. All I can think of is to do $\displaystyle Y-(Y\cap A)$ then but I don't see how that will help.

Re: Y is a subspace of a X which is a topology

Ok, I think I have it.

$\displaystyle A=Y\cap U$ where U is some closed set in X. Now, A is the intersection of two closed sets in X so A is closed in X.

Re: Y is a subspace of a X which is a topology

Quote:

Originally Posted by

**dwsmith** Ok, I think I have it.

$\displaystyle A=Y\cap U$ where U is some closed set in X. Now, A is the intersection of two closed sets in X so A is closed in X.

Correct.